Hamiltonian

canonical momentum

\[p = \frac{\partial \mathcal{L}}{\partial \dot{q}}\] \[H(p,q) = \sum_i p_i\dot{q}_i - L(q,\dot{q})\] \[\frac{\text{d}q^i}{\text{d}t}=\frac{\partial H}{\partial p_i}\] \[\frac{\text{d}p_i}{\text{d}t}=-\frac{\partial H}{\partial q^i}\]

An observable is a general function $f(q,p,t)$ on phase space. It’s total change with time can be obtained with the chain rule as

\[\frac{\text{d}f}{\text{d} t} = \frac{\partial f}{\partial t} + \sum_i \frac{\partial f}{\partial q^i}\frac{\partial q^i}{\partial t} + \sum_i\frac{\partial f }{\partial p_i}\frac{\partial p_i}{\partial t}\]

If we want to see how the function changes along the evolution of the system, we can insert back Hamilton’s equations such that

\[\frac{\text{d}f}{\text{d} t} = \frac{\partial f}{\partial t} + \sum_i \frac{\partial f}{\partial q^i}\frac{\partial H}{\partial p_i} - \sum_i\frac{\partial f }{\partial p_i}\frac{\partial H}{\partial q^i}\]

This motivates the introduction of the so-called Poisson brackets

\[\{f,g\} = \sum_i^{3\mathcal{N}} \left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right)\]

Such that the time evolution of any observable $f$ becomes

\[\frac{\text{d}f}{\text{d} t} = \lbrace f, H\rbrace + \frac{\partial f}{\partial t}\]

While this definition might seem a bit abstract and fancy, the Poisson bracket is of very high importance in the geometry of phase space as discussed in this lecture.